A bag contains one rupee, 50 paise and 25 paise coins. The ratio of the number of 1 rupee coins to that of 50 paise coins is ` 5 : 9` and the ratio of the number of 50 paise conins to that of 25 paise coins is ` 2 : 1` . Find the value of the 50 paise coins in the bag if the total value of the bag is Rs. 425.

Options

A. Rs. 254
B. Rs. 180
C. Rs. 78
D. Cannot be determined

Explanation

To solve the problem step by step, we will follow the ratios given and the total value of the coins.

### Step 1: Define Variables
Let:
- \( x \) = number of 1 rupee coins
- \( y \) = number of 50 paise coins
- \( z \) = number of 25 paise coins

### Step 2: Set Up Ratios
According to the problem:
1. The ratio of 1 rupee coins to 50 paise coins is \( 5:9 \).
\[ \frac{x}{y} = \frac{5}{9} \implies x = \frac{5}{9}y \quad \text{(Equation 1)} \]

2. The ratio of 50 paise coins to 25 paise coins is \( 2:1 \).
\[ \frac{y}{z} = \frac{2}{1} \implies z = \frac{1}{2}y \quad \text{(Equation 2)} \]

### Step 3: Total Value of Coins
The total value of the coins is given as Rs. 425. The value contribution from each type of coin is:
- Value from 1 rupee coins = \( x \times 1 = x \)
- Value from 50 paise coins = \( y \times \frac{1}{2} = \frac{y}{2} \)
- Value from 25 paise coins = \( z \times \frac{1}{4} = \frac{z}{4} \)

Thus, the total value equation is:
\[ x + \frac{y}{2} + \frac{z}{4} = 425 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equations into Total Value
Substituting Equation 1 and Equation 2 into Equation 3:
- Substitute \( x = \frac{5}{9}y \)
- Substitute \( z = \frac{1}{2}y \)

The equation becomes:
\[ \frac{5}{9}y + \frac{y}{2} + \frac{\frac{1}{2}y}{4} = 425 \]

### Step 5: Simplify the Equation
Now simplify the equation:
\[ \frac{5}{9}y + \frac{y}{2} + \frac{1}{8}y = 425 \]

To add these fractions, find a common denominator. The least common multiple of 9, 2, and 8 is 72. Rewriting each term:
\[ \frac{5}{9}y = \frac{40}{72}y, \quad \frac{y}{2} = \frac{36}{72}y, \quad \frac{1}{8}y = \frac{9}{72}y \]

Now combine:
\[ \frac{40}{72}y + \frac{36}{72}y + \frac{9}{72}y = 425 \]
\[ \frac{85}{72}y = 425 \]

### Step 6: Solve for \( y \)
Multiply both sides by \( \frac{72}{85} \):
\[ y = 425 \times \frac{72}{85} \]
Calculating this gives:
\[ y = 360 \]

### Step 7: Find the Value of 50 Paise Coins
The value of the 50 paise coins is:
\[ \text{Value} = y \times \frac{1}{2} = 360 \times \frac{1}{2} = 180 \text{ rupees} \]

### Final Answer
The value of the 50 paise coins in the bag is **Rs. 180**.

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